A Simple Discrete Approximation for the Renewal Function

Transcription

1 Busness Sysems Research Vol. 4 No. 1 / March 2013 A Smple Dscree Approxmaon for he Renewal Funcon Alenka Brezavšček Unversy of Marbor, Faculy of Organzaonal Scences, Kranj, Slovena Absrac Background: The renewal funcon s wdely useful n he areas of relably, manenance and spare componen nvenory plannng. Is calculaon reles on he ype of he probably densy funcon of componen falure mes whch can be, regardng he regon of he componen lfeme, modelled eher by he exponenal or by one of he peak-shaped densy funcons. For mos peak-shaped dsrbuon famles he closed form of he renewal funcon s no avalable. Many approxmae soluons can be found n he leraure, bu calculaons are ofen edous. Smple formulas are usually obaned for a lmed range of funcons only. Objecves: We propose a new approach for evaluaon of he renewal funcon by he use of a smple dscree approxmaon mehod, applcable o any probably densy funcon. Mehods/Approach: The approxmaon s based on he well known renewal equaon. Resuls: The usefulness s proved hrough some numercal resuls usng he normal, lognormal, Webull and gamma densy funcons. The accuracy s analysed usng he normal densy funcon. Conclusons: The approxmaon proposed enables smple and farly accurae calculaon of he renewal funcon rrespecve of he ype of he probably densy funcon. I s especally applcable o he peak-shaped densy funcons when he analycal soluon hardly ever exss. Keywords: componen, falure mes, probably densy funcon, random falures, wear-ou falures, renewal funcon, approxmaon JEL classfcaon: C02, C44 Paper ype: Research arcle Receved: 21, Sepember, 2012 Revsed: 29, November, 2012 Acceped: 24, December, 2012 Caon: Brezavšček, A. (2013). A Smple Dscree Approxmaon for he Renewal Funcon, Busness Sysems Research, Vol. 4, No. 1, pp DOI: /bsrj Inroducon The renewal funcon H () plays an mporan role n he areas of relably, manenance and spare pars nvenory plannng (see e.g. Shekh and Younas, 1985; Barlow, Proschan, and Huner, 1996; Gersbakh, 2000; Brezavšček, 2011). The calculaon of H () relyng on he ype of he probably densy funcon of ner-renewal mes. In he relably and mananably applcaons hs funcon s gven by he probably densy funcon of componen falure mes, usually denoed by he symbol f(). However, s no possble o oban H () analycally for any ype of he funcon f(). For mos peak-shaped dsrbuon famles he closed form of H () s no obanable. Many dfferen approaches have been developed n he leraure o approxmae or numercally compue he renewal funcon. Chaudhry (1995), Garg and Kalagnanam (1998), Cu and Xe (2003), Hu (2006) or Pols and Kouras (2006) gve a comprehensve revew of he approaches for compung he renewal funcon. They found ou ha here are many 65

2 Busness Sysems Research Vol. 4 No. 1 / March 2013 mehods o approxmae he renewal funcon such as he exended cubc-splnng algorhm, he generang funcon algorhm, and power seres expanson (e.g. Robnson, 1997). The frs mehod s no smple o mplemen, he second mehod s me consumng for large values of me, whle he hrd mehod s dsrbuon specfc. Therefore, a smple and general approxmaon for calculang he renewal funcon s desred. The approxmaon should mee he requremens of smplcy (.e. could be used whou a need of furher numercal compuaon), accuracy (.e. should enable farly accurae esmaon of H () for suffcenly large ), and applcably (.e. should be applcable for more dsrbuon famles raher han a specfc dsrbuon). Though some effors have been made o develop such approxmaons (e.g. Smenk and Dekker, 1990; Ayhan, Lmon-Robles and Worman, 1999; Bebbngon, Davydov and Zks, 2007; Jang, 2010) appears ha smple approxmae formulas are obaned for a lmed range of funcons f() only. In hs paper we develop a dscree approxmaon whch enables a smple calculaon of he renewal funcon rrespecve of he ype of he componen falure me dsrbuon funcon. The proposed approxmaon s especally useful for he peak-shaped densy funcons when he analycal soluon of he renewal funcon hardly ever exss. The usefulness of he approxmaon s proved hrough some numercal resuls consderng he normal, lognormal, Webull and gamma densy funcons, whle he accuracy of he approxmaon s analyzed usng he normal densy funcon. Theorecal framework Probably densy funcons of componen falure mes Techncal sysems such as ndusral sysems conss of a number of componens of dfferen ypes. Durng he sysem operaon s componens fals. Tme o falure of a componen of a gven ype s a random varable dsrbued accordng o he probably densy funcon f(). The behavour of he funcon f() depends on he regon of he componen lfeme. From he relably heory s known ha he lfeme of a componen can be roughly dvded no hree regons: he regon of early falures, he regon of random falures (also called he regon of he normal operaon) and he regon of wear-ou falures. Early falures are usually deeced and elmnaed by screenng or burn-n ess before he componens are pu no operaon. The operang perod of he componen hus comprses only he regon of random falures and he regon of wear-ou falures. Random falures are manly due o nheren slowacng defecs n he componen, or o sudden excessve loadng. The man falure mechansm n he wear-ou regon s deeroraon of he componen maerals. The regon of random falures s characerzed by monooncally decreasng funcon f(), whle n he regon of wear-ou falures he dsrbuon of componen falure mes follows a peak-shaped curve. In Fgure 1, he general form of he funcon f() durng he componen lfeme s presened. To he sde, he nsananeous falure rae () s also shown. I can be seen ha durng he regon of random falures () has approxmaely consan value, smply called he falure rae. Durng he wear-ou regon, he nsananeous falure rae () ncreases wh me. The renewal process and he renewal funcon When he sysem durng s operaon fals, a faled componen needs o be replaced by a new one as soon as possble. If he replacemen me s neglgble, he process of consecuve correcve replacemens of a parcular componen can be modelled by an ordnary renewal process. A realzaon of a renewal process consss of a seres of pon evens (renewals) occurrng sngly n me and compleely randomly. When all ner-renewal mes (n our case mes o componen falure) are ndependen dencally dsrbued random varables, he renewal process s called ordnary (e.g. Cox, 1970; Nakagawa, 2011). 66

3 Busness Sysems Research Vol. 4 No. 1 / March 2013 Fgure 1 The Form of he Probably Densy Funcon f() of Componen Falure Tmes and he Insananeous Falure Rae () Source: Auhor s llusraon The essenal characersc of an ordnary renewal process s he renewal funcon H (), defned as he expeced number of renewals of a sngle componen n he nerval (0, ) (e.g. Cox, 1970; Bechel, 2006; Nakagawa, 2011): H( ) E N( ) N represens he number of componen renewals durng he nerval (0, ) where (). The renewal funcon H () can be obaned usng o he equaon (e.g. Cox, 1970; Bechel, 2006; Nakagawa, 2011): (1) H( ) F ( ) r1 r wh F () 1 0 and F1 ( ) F( ) f ( x) dx. The symbol Fr () n (1) represens he r-fold convoluon negral of he cumulave dsrbuon funcon () 0 F : r1, r 2 0 F ( ) F ( x) df( x) r A smple soluon of (1) s avalable for some specfc ypes of f() only. 67

4 Busness Sysems Research Vol. 4 No. 1 / March 2013 Theorecally, he calculaon of he renewal funcon H () accordng o (1) can be smplfed usng he Laplace ransformaons because here s a smple relaon beween he Laplace ransform of a convoluon and he Laplace ransforms of he wo funcons beng convolued (e.g. Cox, 1970; Bechel, 2006). * f Takng no accoun he relaons L f() f () s, * () s L F (), s r f * () s L Fr (), he Laplace ransform s from (1) n he followng form: H * () s of he renewal funcon s easly derved * H () s * 1 f ( s) * s 1 f ( s) (2) The renewal funcon H () s hen obaned by he nverson of he expresson (2): 1 * H( ) L H ( s) (3) Unforunaely, when he funcon f() belongs o one of he peak-shaped dsrbuon famles, f * () s s no always avalable. Even f f * () s s avalable he exac nverson of H * () s n a smple form can hardly ever be obaned. Anoher presenaon of he renewal funcon s provded by he so called renewal equaon (e.g. Tjms, 2003; Nakagawa, 2011): 0 H( ) F( ) H x f x dx (4) Calculaon of he renewal funcon for dfferen ypes of f() I s clear ha he calculaon of H () reles on he ype of he probably densy funcon f() of componen falure mes. I follows from Fgure 1 ha an approprae mahemacal f n he regon of random falures s he exponenal densy funcon, whle n f has go a peak-shaped form whch s usually model for () he regon of wear-ou falures he funcon () modelled by he normal, lognormal, Webull or gamma densy funcon (see e.g. O'Connor, 2011). We shall brefly dscuss he properes of hese ypes of densy funcons from he vewpon of he renewal funcon calculaon. Exponenal f() The exponenal densy funcon s f () e 0, 0 f he where he parameer represens he consan falure rae. For hs ype of () calculaon of H () s rval. The analycal soluon can be smply derved usng he Laplace ransform of f () and equaons (2) and (3). We oban H(). Normal f() The normal densy funcon s gven by he expresson 68

5 Busness Sysems Research Vol. 4 No. 1 / March f () e,, 0 2 wh he parameers mean and sandard devaon. The normal densy funcon, beng defned n he nerval (, ), s an approprae mahemacal model for f() only f / 1. In pracce, we can assume ha hs condon s fulflled when / 3. The negave al of he normal curve s hen a mos If / 3, a runcaed normal dsrbuon can be used (see e.g. Johnson e al., 1994; Koegoda and Rosso, 1997). In he case of normal f() he exac nverson of (2) s no obanable, and he analycal soluon for H () does no exs. However, as he r-fold convoluon of he normal dsrbuon funcon F () wh he parameers and s also a normal dsrbuon funcon wh he parameers r and smple. Lognormal f() The lognormal densy funcon s r, he numercal calculaon of H () accordng o (1) s raher 1 ln f () e 0, 0, 0 2 where and are mean and sandard devaon of ln. f s lognormal he exac nverson of (2) s no obanable, so he analycal When () soluon for H () does no exs. Unforunaely, he numercal calculaon s no rval because he closed form of Fr () s no drecly obanable. Some approxmae formulas can be found n he leraure bu calculaons are prey complex (see e.g. Barouch and Kaufman, 1976; Romeo, Da Cosa and Bardou, 2003; Lam and Le-Ngoc, 2006). Webull f() The wo-parameer 1 Webull densy funcon s f () 1 e 0, 0, 0 where s he shape parameer, and s he scale parameer. If he shape parameer s equal o 1, he Webull densy funcon becomes exponenal, and he calculaon of he renewal funcon s rval. When 1, he Webull densy funcon follows a peak-shaped form. In such a case he closed form of Fr () s no avalable. Consequenly, he numercal calculaon of H () s que edous (see e.g. Jang, 2008). An 1 In relably heory, he hree parameer Webull densy funcon s also used. The hrd parameer,, s he locaon parameer. When 0 he densy funcon sars a me 0. 69

6 Busness Sysems Research Vol. 4 No. 1 / March 2013 exhausve overvew of dfferen mehods for numercal calculaon of he renewal funcon when underlyng dsrbuon s Webull can be found n Rnne (2009). Gamma f() The wo-parameer 2 gamma densy funcon s 1 f () ( ) 1 e 0, 0, 0 where represens he shape parameer, s he scale parameer 3, and (.) denoes he gamma funcon 4. If 1, he gamma densy funcon s equal o he exponenal densy funcon, and he calculaon of H () s rval. Smlar o he Webull densy funcon, he gamma densy funcon follows a peak-shaped form when 1. The analycal soluon for H () only exss for some neger values of (e.g., 2 or 3). The numercal calculaon of H () accordng o (1) s smple for any value of. Namely, he r-fold convoluon of he gamma dsrbuon funcon wh he parameers and s also a gamma dsrbuon funcon wh he parameers r and. f s exponenal We can conclude ha he analycal soluon for H () s rval when () (.e. componens operae n he regon of random falures) whle he closed form of H () s hardly ever avalable when he underlyng densy funcon s a peak-shaped (.e. he componens operae n he regon of wear-ou falures). The dervaon of an approxmae soluon of H () for a peak-shaped f() has araced he aenon of many auhors, bu smple formulas are obaned for a lmed range of funcons f() only. A more general approxmae soluon for H () would be mos desrable. The approxmae soluon for he renewal funcon Our am s o derve an approxmae soluon for evaluang he renewal funcon whch should mee he followng requremens: Mahemacal operaons nvolved are smple. Sasfacory accuracy s acheved even for suffcenly large value of me. The soluon s useful for any ype of he probably densy funcon of componen falure mes. Especally s applcable o he peak-shaped probably densy funcons whch are useful o descrbe he componen falure me dsrbuon n he wear-ou regon. 2 In relably heory, he hree parameer gamma densy funcon s also used. The hrd parameer,, s he locaon parameer. When 0 he densy funcon sars a me 0. 3 When s neger he gamma densy funcon becomes he Erlang densy funcon. When he shape parameer s 2 (β s any neger) and he scale parameer s equal o 2 he gamma densy funcon becomes he Chsquare densy funcon. 4 x1 z x z e dz ( ) 0 70

7 Busness Sysems Research Vol. 4 No. 1 / March 2013 Accordng o our knowledge, such a soluon has no been provded n he leraure ll now. As a bass for he dervaon of our approxmaon he recursve negral equaon (4) s used. The man dea of he approxmaon s a dscrezaon of he connuous me nerval (0, T ) by s dvson o a number of subnervals of lengh h, as shown n Fgure 2. The lengh h of subnervals whn (0, T ) should be shor enough o ensure ha he probably of occurrng more han one componen falure durng h s neglgble. Fgure 2 Dscrezaon of he Inerval (0, T ) Source: Auhor s llusraon If he value h s shor enough, we can assume ha componen renewals durng (0, T ) occur only n he me pons, 1, 2,, T assumpon our dscree approxmaon of (4) reads as follows: H( T) p H( T ) p T 1,2, H(0) 0 T T 1 1 (5) wh he probably p. Consderng hs We defne he probably p n (5) by he negral p f () d, 1,2,, 1 1 p 1 Ths defnon mples ha he values of dscree and connuous cumulave dsrbuon funcons of ner-renewal mes are equal snce: T T p f ( ) d f ( ) d F( T) (6) T Consderng he equaons (5) and (6) we ge he followng algorhm for esmang he value HT ( ) of he renewal funcon n an arbrary me T : T H( T) F( T) H( T ) f ( ) d T 1,2, 1 1 (7) wh he nal condon H(0) 0. Snce our approxmaon (7) nvolves he assumpon ha he renewals whch can acually occur anywhere whn he nerval (0, T ) occur exacly a he me pons, 1, 2,, T, he value HT ( ), calculaed accordng o (7), s oo low. The error nroduced n such a way can 71

8 Busness Sysems Research Vol. 4 No. 1 / March 2013 be dmnshed by choosng suffcenly small value of h. When h 0 he value HT ( ), calculaed accordng o (7), converges o he exac value of he renewal funcon. However, decreasng of h wll mprove he accuracy of our approxmaon (7) bu also ncrease he number of recursve calculaons. Therefore, he lengh h of subnervals beween wo successve me pons, 1, 2,, T, s approprae, when s shorenng does no change he value of HT ( ) over accepable lms. Because he mahemacal operaons n (7) are smple, he renewal funcon can be calculaed easly for any value of me T and regardless of he ype of he funcon f(). Smlar approxmaons, derved n a dfferen way, have been proposed n Jardne (1973), Jardne and Tsang (2006) and van Noorwjk and van der Wede (2008). Numercal resuls To prove he usefulness of he approxmaon (7) some numercal resuls are gven n Table 1. In he calculaons, he normal, lognormal, Webull and gamma densy funcons are consdered. Resuls from Table 1 prove ha our approxmaon (7) s applcable o all ypes of densy funcons whch are wdely used n relably sudes o descrbe he funcon f() n he wearou regon. Alhough all of densy funcons consdered are of peak-shaped form, he calculaed values of HT ( ) canno be compared drecly. In he case of he normal and he gamma densy funcons he values HT ( ), calculaed accordng (7), can be compared wh he correspondng values, calculaed accordng o (1). We can brefly conclude ha our approxmaon gves farly accurae resuls even for comparavely large values of me. Table 1 Applcably of he Approxmaon (7) o Dfferen Peak-Shaped Probably Densy Funcons, Wdely Used n he Relably Applcaons Normal pdf Lognormal pdf ' 5 ' H () calculaed accordng o (1) H () calculaed accordng o (7) H calculaed () accordng o (1) H calculaed () accordng o (7) Weblull pdf H () calculaed accordng o (1) H () calculaed accordng o (7) Gamma pdf H () calculaed accordng o (1) H () calculaed accordng o (7) Source: Auhor s Calculaons E The accuracy of he approxmaon (7) depends on he lengh h of subnervals beween T (see Fgure 2). To analyze he effec of shorenng wo successve me pons, 1, 2,, of h o he accuracy of he calculaed values we use he normal densy funcon. In order o 72

9 Busness Sysems Research Vol. 4 No. 1 / March exclude he possble effec of he negave al we presen he resuls for he rao 4 HT, calculaed accordng o (1) 5, and he correspondng We compare he values of ( ) values, calculaed accordng o (7). In calculaons accordng o (7) dfferen values of h are consdered. Resuls for he me span up o 3 are shown n Fgure 3. Fgure 3 The nfluence of he lengh h of subnervals beween wo successve me pons, on he accuracy of he approxmaon (7), 1,2,, T Source: Auhor s llusraon I can be seen from Fgure 3 ha for T 1.2 all he curves overlap. Ths proves ha for farly shor mes our approxmaon (7) provdes accurae resuls rrespecve of he value of h. When T ncreases he error of HT ( ) calculaed usng (7) ncreases. However, we can see ha shorenng of h dmnsh he error effcenly. When h 30 he error does no exceed 1% for mes beween and 2. For smaller values of h sll beer accuracy s obaned. For h he error does no exceed 1% even for T 3. example, f 50 Concluson The renewal funcon s an mporan characersc, needed n he areas of relably analyses, manenance opmzaon and spare componens nvenory plannng. Is calculaon depends on he form of he probably densy funcon of falure mes durng he regon of componen operaon. In he regon of random falures, he calculaon of he renewal funcon s rval because he probably densy funcon of componen falure mes s 5 The approxmaon 6 H( T) F ( T) s used. The maxmum error does no exceed 3E-6. r1 r 73

10 Busness Sysems Research Vol. 4 No. 1 / March 2013 descrbed by he exponenal densy funcon. In he regon of wear-ou falures, he probably densy funcon of componen falure mes exhbs a peak-shaped form. In ha case, he calculaon of he renewal funcon s n general dffcul because he analycal soluons hardly ever exs. The dervaon of an approxmae soluon of he renewal funcon for a peak-shaped falure me dsrbuon has araced he aenon of many auhors, bu smple approxmae formulas are obaned for a lmed range of funcons only. In hs paper, we have proposed a dscree approxmaon for esmang he value of he renewal funcon. The approxmaon s derved from he so called renewal equaon. Due o smple mahemacal operaons nvolved, enables a smple and farly accurae calculaon of he renewal funcon for any ype of he probably densy funcon of componen falure mes. I s especally applcable o he peak-shaped densy funcons when he analycal soluons hardly ever exs. To prove he usefulness of he approxmaon some numercal resuls are gven consderng he normal, lognormal, Webull and gamma probably densy funcons. I s also shown ha sasfacory accuracy of he approxmaon can be acheved by dvson of he me nerval no suffcenly shor subnervals. Snce he approxmaon proposed mees he requremens of he smplcy as well as he applcably o dfferen ypes of peak-shaped densy funcons, he praccal value of he approxmaon s sgnfcan. The approxmaon s useful everywhere n he relably analyss and he manenance polcy opmzaon where he renewal funcon needs o be evaluaed. The man lmaon of our approxmaon s he followng: The accuracy of our approxmaon depends on he lengh h of subnervals beween wo successve me pons, where smaller h ensures beer accuracy (see Fgure 2). However, he negave effec of he decreasng of h resuls n he ncreasng of he number of recursve calculaons. Consequenly, when someone wans o calculae very accurae esmaon of he renewal funcon n a very large value of me (.e. ), our approxmaon becomes me consumng, and herefore probably useless. However, has been shown ha whn he accepable lms of calculang me he error less han 1% n he me span up o 3 can be acheved. In our opnon, for praccal purposes hs s good enough, even because a smple asympoc formula for he renewal funcon (Cox, 1970; Bechel, 2006) can be used for larger values of me. References 1. Ayhan H., Lmon-Robles, J., Worman M. A. (1999), An approach for compung gh numercal bounds on renewal funcons, IEEE Transacons on Relably, Vol. 48 No. 2, pp Barlow, R. E., Proschan, F., Huner, L. C. (1996). Mahemacal Theory of Relably, Phladelpha, SIAM. 3. Barouch, E., Kaufman, G. M. (1976), On Sums of Lognormal Random Varables. Workng paper, Alfred P. Sloan School of Managemen, Cambrdge, Massachuses, avalable a hp://dspace.m.edu/bsream/handle/1721.1/48703/onsumsoflognorma00baro.pdf / (10 June 2011). 4. Bebbngon, M., Davydov, Y., Zks, R. (2007), Esmang he renewal funcon when he second momen s nfne, Sochasc Models, Vol. 23 No.1, pp Bechel, F. (2006). Sochasc Processes n Scence, Engneerng And Fnance, Boca Raon, Chapman & Hall/CRC. 6. Brezavšček, A. (2011), Smple Sochasc Model for Plannng he Invenory of Spare Componens Subjec o Wear-ou, Organzacja, Vol. 44 No. 4, pp Chaudhry, M. L. (1995), On compuaons of he mean and varance of he number of renewals: a unfed approach, The Journal of he Operaonal Research Socey, Vol. 46 No. 11, pp Cox, D. R. (1970). Renewal Theory, London & Colcheser: Mehuen. 74

11 Busness Sysems Research Vol. 4 No. 1 / March Cu, L., Xe, M. (2003), Some normal approxmaons for renewal funcon of large Webull shape parameer, Communcaons n Sascs - Smulaon and Compuaon, Vol. 32 No. 1, pp Garg, A., Kalagnanam, J. R. (1998), Approxmaons for he renewal funcon, IEEE Transacons on Relably, Vol. 47 No. 1, pp Gersbakh, I. (2000). Relably Theory, Wh Applcaons o Prevenve Manenance, Berln: Sprnger Verglag. 12. Hu, X. (2006), Approxmaon of paral dsrbuon n renewal funcon calculaon, Compuaonal Sascs & Daa Analyss, Vol. 50 No. 6, pp Jardne, A. K. S. (1973). Manenance, Replacemen, and Relably, London, Pman. 14. Jardne, A. K. S., Tsang, A. H. C. (2006). Manenance, Replacemen, and Relably: Theory and Applcaons, Boca Raon, CRC/Taylor & Francs. 15. Jang, R. (2008), A Gamma normal seres runcaon approxmaon for compung he Webull renewal funcon, Relably Engneerng & Sysem Safey, Vol. 93 No. 4, pp Jang, R. (2010), A smple approxmaon for he renewal funcon wh an ncreasng falure rae, Relably Engneerng & Sysem Safey, Vol. 95 No. 9, pp Johnson, N. L, Koz, S., Balakrshnan, N. (1994), Connuous Unvarae Dsrbuons, Volumes I and II, 2nd. Ed., New York: John Wley and Sons. 18. Koegoda, N. T., Rosso, R. (1997). Sascs, Probably, and Relably for Cvl and Envronmenal Engneers, New York: McGraw-Hll. 19. Lam, C. L. J., Le-Ngoc, T. (2006), Esmaon of ypcal sum of lognormal random varables usng log shfed gamma approxmaon, IEEE Communcaons Leers, Vol. 10 No. 4, pp Nakagawa, T. (2011). Sochasc Processes: wh Applcaons o Relably Theory, London: Sprnger-Verlag. 21. O'Connor, A. N. (2011). Probably Dsrbuons Used n Relably Engneerng, Maryland: RIAC. 22. Pols, K., Kouras, M. V. (2006), Some new bounds for he renewal funcon, Probably n he Engneerng and Informaonal Scences, Vol. 20 No. 2, pp Rnne, H. (2009). The Webull Dsrbuon: A Handbook, New York: CRC Press, Taylor & Francs Group. 24. Robnson, N. I. (1997), Renewal funcons as seres, Sochasc Models, Vol. 13 No. 3, pp Romeo, M., Da Cosa, V., Bardou, F. (2003), Broad dsrbuon effecs n sums of lognormal random varables, The European Physcal Journal B - Condensed Maer and Complex Sysems, Vol. 32 No. 4, pp Shekh, A. K., Younas, M. (1985), Renewal Models n Relably Engneerng, n Deopker, P. E. (Ed.), Falure and Prevenon and Relably, ASME, pp Smenk, E., Dekker, R. (1990), A smple approxmaon o he renewal funcon, IEEE Transacons on Relably, Vol. 39 No. 1, pp Tjms, H. C. (2003). A Frs Course n Sochasc Models, Chcheser: John Wley & Sons. 29. van Noorwjk, J. M., van der Wede, J. A. M. (2008), Applcaons o connuous-me processes of compuaonal echnques for dscree-me renewal processes, Relably Engneerng & Sysem Safey, Vol. 93 No. 12, pp Abou he auhor Alenka Brezavšček (1967) s an asssan professor a he Faculy of Organzaonal Scences, Unversy of Marbor, Slovena. Her research felds are sochasc processes, sysem relably and avalably, and nformaon sysem secury. Auhor can be reached a alenka.brezavscek@fov.un-mb.s 75